Annual per capita consumption of milk is 21.6 gallons (Statistical Abstract of the United States: 2006). Being from the Midwest, you believe milk consumption is higher there and wish to support your opinion. A sample of 16 individuals from the mid western town of Webster City showed a sample mean annual consumption of 24.1 gallons with a standard deviation of s = 4.8.

1. What is a point estimate of the difference between mean annual consumption in Webster City and the national mean?

2. At = .05, test for a significant difference by completing the following.
Calculate the value of the test statistic (to 2 decimals).

3. The p-value is?

1. To find the point estimate of the difference between the mean annual consumption in Webster City and the national mean, we subtract the national mean from the sample mean.

Point estimate = Sample mean - National mean
Point estimate = 24.1 gallons - 21.6 gallons
Point estimate = 2.5 gallons

Therefore, the point estimate of the difference between mean annual consumption in Webster City and the national mean is 2.5 gallons.

2. To test for a significant difference at α = 0.05, we can conduct a t-test using the given information.

Step 1: Set up hypotheses:
Null hypothesis (H0): The mean annual consumption in Webster City is not significantly different from the national mean.
Alternative hypothesis (H1): The mean annual consumption in Webster City is significantly different from the national mean.

Step 2: Calculate the test statistic:
The formula for the t-test statstic when standard deviation is known is:
t = (sample mean - population mean) / (standard deviation / sqrt(sample size))

Using the given information:
Sample mean (x̄) = 24.1 gallons
Population mean (μ) = 21.6 gallons
Standard deviation (s) = 4.8 gallons
Sample size (n) = 16

t = (24.1 - 21.6) / (4.8 / sqrt(16))
t = 2.5 / (4.8 / 4)
t ≈ 2.5 / 1.2
t ≈ 2.08 (to two decimal places)

Therefore, the value of the test statistic is approximately 2.08.

3. To find the p-value, we will use the t-distribution with degrees of freedom equal to the sample size minus 1 (n - 1).

Using the t-distribution table or a statistical software, we find the p-value associated with a t-statistic of 2.08.

Assuming a two-tailed test because we are testing for a significant difference in either direction, the p-value is the probability of obtaining a t-statistic greater than or equal to 2.08 or less than or equal to -2.08.

Looking up the p-value for a t-statistic of 2.08 in a t-distribution table, we find that p-value ≈ 0.051 (to three decimal places).

Therefore, the p-value is approximately 0.051.